# 13.1: Inverse Variation

Difficulty Level: At Grade Created by: CK-12

This activity is intended to supplement Algebra I, Chapter 12, Lesson 1.

## Part 1 - Enter the Data

Enter the data from the table into lists.

Press STAT ENTER. Enter the x\begin{align*}x\end{align*} column in L1\begin{align*}L1\end{align*} and the y\begin{align*}y\end{align*} column in L2\begin{align*}L2\end{align*} as shown.

x\begin{align*}x\end{align*} y\begin{align*}y\end{align*}
1 24\begin{align*} 24\end{align*}
2 12\begin{align*}12\end{align*}
3 8\begin{align*}8\end{align*}
4 6\begin{align*}6\end{align*}
5 4.8\begin{align*}4.8\end{align*}
6 4\begin{align*}4\end{align*}

Press Y=\begin{align*}Y=\end{align*}, and select Plot1.

Press ENTER to turn the plot On. Select scatter as the type of plot, L1\begin{align*}L1\end{align*} for the Xlist, and L2\begin{align*}L2\end{align*} for the Ylist.

Press WINDOW. Set the window to the following:

Xmin=0, Xmax=10, Xscl=2\begin{align*}Xmin = 0,\ Xmax = 10,\ Xscl = 2\end{align*}

Ymin=0, Ymax=25, Yscl=5\begin{align*}Ymin = 0,\ Ymax = 25,\ Yscl = 5\end{align*}

Press GRAPH.

## Part 2 - Questions

• How would you describe the relationship between x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} by examining this data?

Press STAT ENTER to return to the lists.

• What relationships can you see by examining the numbers in the lists?
• What is the product of each pair of numbers?

Arrow to the top of L3\begin{align*}L3\end{align*}. Enter a formula to multiply the entries in L1\begin{align*}L1\end{align*} by the entries in L2\begin{align*}L2\end{align*}. Press 2nd [L1]\begin{align*}2^{nd}\ [L1]\end{align*} for L1\begin{align*}L1\end{align*} and press 2nd [L2]\begin{align*}2^{nd}\ [L2]\end{align*} for L2\begin{align*}L2\end{align*}. L3=L1L2\begin{align*}L3 = L1*L2\end{align*}

Press ENTER to execute the formula. The product in each case is 24\begin{align*}24\end{align*}. So, L1L2=24\begin{align*}L1 \cdot L2 = 24\end{align*} or xy=24\begin{align*}x \cdot y = 24\end{align*}. This relationship, when x\begin{align*}x\end{align*} and y\begin{align*}y\end{align*} have a constant product, is called “inverse variation.”

• What type of situation might this be a formula for?

Solve the equation xy=24\begin{align*}x \cdot y = 24\end{align*} for y\begin{align*}y\end{align*}. Press Y=\begin{align*}Y=\end{align*}. Enter the equation into Y1\begin{align*}Y1\end{align*}.

• What is your equation?

Press GRAPH.

• What other information can you find from the graph of the equation that you could not gather from the plot?
• Does this graph appear to be a function? Explain.

Press 2nd\begin{align*}2^{nd}\end{align*} [TABLE] to examine the function table.

• What is happening when x=0\begin{align*}x = 0\end{align*}? Why?

Arrow up to the negative x\begin{align*}x-\end{align*}values in the table.

• What do you notice about the y\begin{align*}y-\end{align*}values?
• Why does this occur?
• What do you think the graph of your equation looks like to the left of the y\begin{align*}y-\end{align*}axis?

Press WINDOW. Set the window as shown to examine the graph when x\begin{align*}x\end{align*} is negative.

Press GRAPH.

• What appears to be happening when x=0\begin{align*}x = 0\end{align*}?
• Why does the graph of the equation not appear in Quadrants II or IV?
• Do you think an inverse variation can ever be found in Quadrants II or IV? Why?
• Does this graph appear to be a function now? Explain.

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